West Virginia came into being because the western counties setup a rump government headed up by Pierpont. The convention to setup the West Virginia government had persons representing counties that they did even live in. When the vote came on whether the specified counties were to secede from Virginia, the western counties were occupied by Union troops. The vote totals were substantially less the vote totals held 3 years previously when Virginia voted to secede. I believe in some counties, the vote totals were in the hundreds. Of the counties that presently make up the state of West Virginia, about half voted to secede when Virginia seceded.
Lee’s quote is quite accurate about Virginia trying to hold the Union together. During the secession convention, Virginia voted not to secede (I think the date was Apri 3, 1861) and sent a delegation to Washington. Lincoln refused to meet the delegation. After Lincoln called up troops to invade the South, Virginia refused to be a party to the invasion and seceded.
The northern parts of West Virginia were settled from PA and I think the push to break away from Virginia came mainly from those areas, not the areas settled from the eastern parts of Virginia.